Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(4897\)\(\medspace = 59 \cdot 83 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.4897.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Determinant: | 1.4897.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.1.4897.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - x^{3} + x^{2} + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 2 + 12\cdot 13 + 3\cdot 13^{2} + 7\cdot 13^{3} + 9\cdot 13^{4} +O(13^{5})\) |
$r_{ 2 }$ | $=$ | \( 4 a + 3 + 5 a\cdot 13 + 6\cdot 13^{2} + \left(11 a + 6\right)\cdot 13^{3} + \left(4 a + 4\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 3 }$ | $=$ | \( a + 7 + \left(5 a + 10\right)\cdot 13 + \left(8 a + 5\right)\cdot 13^{2} + \left(8 a + 10\right)\cdot 13^{3} + \left(6 a + 7\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 4 }$ | $=$ | \( 9 a + 7 + \left(7 a + 1\right)\cdot 13 + \left(12 a + 1\right)\cdot 13^{2} + \left(a + 4\right)\cdot 13^{3} + \left(8 a + 11\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 5 }$ | $=$ | \( 12 a + 8 + \left(7 a + 1\right)\cdot 13 + \left(4 a + 9\right)\cdot 13^{2} + \left(4 a + 10\right)\cdot 13^{3} + \left(6 a + 5\right)\cdot 13^{4} +O(13^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 5 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 5 }$ | Character value | Complex conjugation |
$1$ | $1$ | $()$ | $4$ | |
$10$ | $2$ | $(1,2)$ | $2$ | |
$15$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
$20$ | $3$ | $(1,2,3)$ | $1$ | |
$30$ | $4$ | $(1,2,3,4)$ | $0$ | |
$24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
$20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |